Many beyond the Standard Model (BSM) scenarios involve Higgs couplings to additional electroweak fields. It is well established that these new fields may modify Higgs gamma-gamma and gamma-Z decays at one-loop. However, one unexplored aspect of such scenarios is that by electroweak symmetry one should also expect modifications to the Higgs Z-Z coupling at one-loop and, more generally, modifications to Higgs production and decay channels beyond tree-level. In this talk I will discuss the full BSM modified electroweak corrections to associated Higgs production at both the LHC and a future lepton collider in two simple SM extensions, and discuss prospects for their determination at the LHC and future lepton colliders.